Stolarsky mean

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In mathematics, the Stolarsky mean of two positive real numbers xy is defined as:

It is derived from the mean value theorem, which states that a secant line, cutting the graph of a differentiable function at and , has the same slope as a line tangent to the graph at some point in the interval .

The Stolarsky mean is obtained by

when choosing .

Special cases[edit]

  • is the minimum.
  • is the geometric mean.
  • is the logarithmic mean. It can be obtained from the mean value theorem by choosing .
  • is the power mean with exponent .
  • is the identric mean. It can be obtained from the mean value theorem by choosing .
  • is the arithmetic mean.
  • is a connection to the quadratic mean and the geometric mean.
  • is the maximum.

Generalizations[edit]

One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the nth derivative. One obtains

for .

See also[edit]

References[edit]